188 ANALYTICAL MECHANICS PROBLEMS. 1. Show that the dimensions of work and kinetic energy are the same. 2. A body of 50 gm. mass starts from the top of an inclined plane 10 m. long, and arrives at the bottom with a velocity of 300 —-. Find the aver- sec. age frictional force. The angle of elevation of the plane is 30°. 3. A body of 100 gm. mass, which is projected up an inclined plane, cm. arrives at the top of the plane with a velocity of 150—1. Find the sec. velocity of projection, supposing the frictional force to be constant and equal to 10,000 dynes. The length of the plane is 5 inches, and the angle of elevation is 30°. 4. A bullet enters a plank with a velocity of 1500 —-, and leaves it sec. with a velocity of 1350 —- - How many such planks can the bullet, sec. penetrate? 5. In the preceding problem find the average resisting force which the planks offer. The bullet weighs \ ounce. 6. A catapult, which consists of an elastic string 15 cm. long, with its ends tied to the prongs of a forked piece of wood, is used to throw stones. What velocity will it give to a stone of 5 gm. mass when stretched to twice its natural length. The modulus of elasticity of the string is 2 pounds. 7. The kinetic energy acquired by a weight of 750 pounds in falling through a distance of 4 feet is to be absorbed by a helical spring, 5 inches long. Find the modulus of elasticity of the spring so that it will not be compressed more than 1 inch. 8. Having a given size and shape, how will the penetrative power of a bullet depend (a) on its weight, and (b) on its velocity. The resisting force is supposed to be constant. 155. Kinetic Energy of a Rigid Body Rotating About a Fixed Axis. — Suppose the rigid body A, Fig. 101, to rotate about an axis through the point 0, at right angles to the plane of the paper. Consider the kinetic energy of an element of mass dm at a distance r from the axis. If v denotes the velocity of the element and dT its kinetic energy, then dT=%v* dm = | r2co2 dm} [v = rw]